We aim to prove Theorem REF  by utilizing Sion's minimax theorem:
[Sion's minimax theorem [1]}] 
Let \(\mathcal {X}\)  be a convex subset of a linear topological space and \(\mathcal {Y}\)  a compact convex subset of a linear topological space.
If \(f\)  is a real-valued function on \(\mathcal {X}\times \mathcal {Y}\)  with
\(f(x,\cdot )\)  upper semicontinuous and quasi-concave on \(Y\) , for each \(x \in \mathcal {X}\) , and
\(f(\cdot ,y)\)  lower semicontinuous and quasi-convex on \(X\) , for each \(y \in \mathcal {Y}\) ,
then,
\(\inf _{x \in \mathcal {X}} \max _{y \in \mathcal {Y}} f(x,y) = \max _{y \in \mathcal {Y}} \inf _{x \in \mathcal {X}} f(x,y),\) 
